edits

main.tex

@@ -132,7 +132,7 @@
                                   & \noindent\large{Anton Rudenko,$^{\ast}$\textit{$^{a}$}  Konstantin Ladutenko,\textit{$^{b}$}  Sergey Makarov\textit{$^{b}$} and Tatiana E. Itina\textit{$^{a}$$^{b}$}
  
                                     \textit{$^{a}$~Laboratoire Hubert Curien, UMR CNRS 5516, University of Lyon/UJM, 42000, Saint-Etienne, France }
-                                    \textit{$^{b}$~ITMO University, Kronverksiy pr. 49, St. Petersburg, Russia}
+                                    \textit{$^{b}$~ITMO University, Kronverksiy pr. 49, Saint-Petersburg, Russia}
  
                                     } \\%Author names go here instead of "Full name", etc.
  
@@ -147,7 +147,7 @@
   regime of simultaneous excitation of electric and magnetic Mie
   resonances, resulting in an effective transient reconfiguration of
   nanoparticle scattering properties. However, only homogeneous plasma
-  generation was previously considered in the photo-excited
+  generation was previously considered in a photo-excited
   nanoparticle, remaining unexplored any effects related to the
   plasma-induced optical inhomogeneities. Here we examine these
   effects by using 3D numerical modeling of coupled electrodynamic and
@@ -215,8 +215,8 @@ modulation~\cite{iyer2015reconfigurable, makarov2015tuning,
 all-dielectric nanoantennas and metasurfaces possess much smaller
 parasitic Joule losses at high intensities as compared with their
 plasmonic counterparts, whereas their nonlinear properties are
-comparable. More importantly, the unique properties of the nonlinear
-all-dielectric nanodevices are due to the existance of both electric and
+comparable. More importantly, the unique properties of nonlinear
+all-dielectric nanodevices are due to the existence of both electric and
 magnetic optical resonances in visible and near IR
 ranges~\cite{kuznetsov2016optically}. For instance, even slight
 variation of dielectric permittivity around optical resonances leads
@@ -253,7 +253,7 @@ symmetry of nanoparticles can be reduced.
 
 In this Letter, we show theoretically that ultra-fast photo-excitation
 in a spherical silicon NP leads to a strongly inhomogeneous EHP
-distribution, as schematically shown in Fig.~\ref{fgr:concept}. To
+distribution, as is shown schematically in Fig.~\ref{fgr:concept}. To
 reveal and analyze this effect, we perform a full-wave numerical
 simulation. We consider an intense femtosecond (\textit{fs}) laser
 pulse to interact with a silicon NP supporting Mie resonances and
@@ -295,9 +295,9 @@ advantage is based on a broad range of optical nonlinearities, strong
 two-photon absorption, as well as a possibility of the photo-induced
 EHP excitation~\cite{leuthold2010nonlinear}. Furthermore, silicon
 nanoantennas demonstrate a sufficiently high damage threshold due to
-the large melting temperature ($\approx 1690$~K), whereas its nonlinear
-optical properties have been extensively studied during last
-decays~\cite{Van1987, Sokolowski2000, leuthold2010nonlinear}. High
+the high melting temperature ($\approx 1690$~K), whereas its nonlinear
+optical properties were extensively studied during the last
+decades~\cite{Van1987, Sokolowski2000, leuthold2010nonlinear}. High
 silicon melting point typically preserves structures formed from this
 material up to the EHP densities on the order of the critical value
 $N_{cr} \approx 5\cdot{10}^{21}$~cm$^{-3}$ \cite{Korfiatis2007}. At
@@ -372,8 +372,8 @@ Gaussian slightly focused beam as follows
 \end{aligned}
 \end{align}
 where $\theta \approx 130$~\textit{fs} is the temporal pulse width at
-the half maximum (FWHM), $t_0$ is a time delay, $w_0 = 3{\mu}m$ is the
-waist beam, $w(z) = {w_0}\sqrt{1+(\frac{z}{z_R})^2}$ is the Gaussian's
+the half maximum (FWHM), $t_0$ is the time delay, $w_0 = 3{\mu}m$ is the
+beam waist, $w(z) = {w_0}\sqrt{1+(\frac{z}{z_R})^2}$ is the Gaussian's
 beam spot size, $\omega = 2{\pi}c/{\lambda}$ is the angular frequency,
 $\lambda = 800$~nm is the laser wavelength in air, $c$ is the speed of
 light, ${z_R} = \frac{\pi{{w_0}^2}n_0}{\lambda}$ is the Rayleigh
@@ -403,7 +403,7 @@ $I=\frac{n}{2}\sqrt{\frac{\epsilon_0}{\mu_0}}\left|\vec{E}\right|^2$
 is the intensity, $\sigma_1 = 1.021\cdot{10}^3$~cm$^{-1}$ and
 $\sigma_2 = 0.1\cdot{10}^{-7}$~cm/W are the one-photon and two-photon
 interband cross-sections \cite{Choi2002, Bristow2007, Derrien2013},
-$N_a = 5\cdot{10}^{22}$~cm$^{-3}$ is the saturation particle density
+$N_a = 5\cdot{10}^{22}$~cm$^{-3}$ is the particle saturation density
 \cite{Derrien2013}, $C = 3.8\cdot{10}^{-31}$~cm$^6$/s is the Auger
 recombination rate \cite{Van1987}, $\tau_{rec} = 6\cdot{10}^{-12}$~s
 is the minimum Auger recombination time \cite{Yoffa1980}, and
@@ -526,13 +526,13 @@ GitHub~\cite{Scattnlay-web} under open source license.
  the intensity is enhanced in the back side of the NP as demonstrated
  in Fig.~\ref{mie-fdtd}(d). In fact, rather similar EHP distributions
  can be obtained by applying Maxwell's equations coupled with the rate
- equation for relatively weak excitation with EHP concentration of
+ equation for a relatively weak excitation with EHP concentration of
  $N_e \approx 10^{20}$~cm$^{-3}$, see Fig.~\ref{mie-fdtd}(e,f). The
  optical properties do not change considerably due to the excitation
  according to (\ref{Index}). Therefore, the excitation processes
  follow the intensity distribution. However, such coincidence was
  achieved under quasi-stationary conditions, after the electric field
- made enough oscillations inside the Si NP. Further on we present
+ made enough oscillations inside the Si NP. Further on, we present
  transient analysis, which reveals much more details.
 
  To achieve a quantitative description for evolution of the EHP
@@ -551,7 +551,7 @@ GitHub~\cite{Scattnlay-web} under open source license.
  be justified. In what follows, we discuss the results of the
  numerical modeling of the temporal evolution of EHP densities and the
  asymmetry factors $G_{N_e}$ for different sizes of Si NP, as shown in
- Fig.~\ref{time-evolution}). Typical change of the permittivity
+ Fig.~\ref{time-evolution}). Typical change in permittivity
  corresponding to each stage is shown in Fig.~\ref{plasma-grid}. It
  reveals the EHP evolution stages during interaction of femtosecond
  laser pulse with the Si NPs.
@@ -596,7 +596,7 @@ GitHub~\cite{Scattnlay-web} under open source license.
  Fig.~\ref{plasma-grid}(f,j). We stress the nonstationary nature of
  E-field pattern at this stage, whereas there is a simultaneous growth
  of the incident pulse amplitude. This leads to a superposition of ED
- near-field pattern with that from the Stage 1, resulting in EHP
+ near-field pattern with that from Stage 1, resulting in EHP
  concentration in the front side of the Si NP. This effect dominates
  for the smallest NP with $R=75$~nm in Fig.~\ref{plasma-grid}(b),
  where ED mode is tuned far away from the resonance (see
@@ -609,7 +609,7 @@ GitHub~\cite{Scattnlay-web} under open source license.
  When the number of optical cycles is large enough
  ($t>20$~\textit{fs}) both ED and MD modes can be exited to the level
  necessary to achieve the stationary intensity pattern corresponding
- to the Mie-based intensity distribution at the \textit{'Stage~3'}
+ to the Mie-based intensity distribution at \textit{'Stage~3'}
  (see Fig.~\ref{plasma-grid}(c,g,k)). The EHP density for the most
  volume of NP is still relatively small to affect the EHP evolution,
  but is already high enough to change the local optical properties,
@@ -623,18 +623,18 @@ GitHub~\cite{Scattnlay-web} under open source license.
  resonant size for $R = 115$~nm the $G_{N_e} < 0$ due to the fact that
  EHP is dominantly localized in the back side of the NP.
 
- In other words, due to a quasi-stationary pumping during the Stage~3
- is superposed with the Stage~1 field pattern, resulting in an
+ In other words, due to a quasi-stationary pumping during Stage~3,
+ it is superposed with Stage~1 field pattern, resulting in an
  additional EHP localized in the front side. This can be seen when
- comparing result from the Mie theory in Fig.~\ref{mie-fdtd}(d) and
- result of full 3D simulation in Fig.~\ref{mie-fdtd}(f). Note that
+ comparing result from the Mie theory in Fig.~\ref{mie-fdtd}(d) with
+ that of full 3D simulation in Fig.~\ref{mie-fdtd}(f). Note that
  pumping of NP significantly changes during a single optical cycle,
- this leads to a large variation of asymmetry factor $G_{N_e}$ at
- first stage. This variation steadily decrease as it goes to Stage~3,
+ this leads to a large variation of asymmetry factor $G_{N_e}$ at the
+ first stage. This variation steadily decreases as it goes to Stage~3,
  as shown in Fig.~\ref{time-evolution}(b,d,f).
 
  To explain this effect, we consider the time evolution of average EHP
- densities $N_e$ in the front and back halves of NP presented in
+ densities $N_e$ in the front and back halves of the NP presented in
  Fig.~\ref{time-evolution}(a,c,e). As soon as the recombination and
  diffusion processes are negligible at \textit{fs} time scale, both
  $N_e^{front}$ and $N_e^{back}$ curves experience monotonous behavior
@@ -642,14 +642,14 @@ GitHub~\cite{Scattnlay-web} under open source license.
  the back halves of NP are separated in space, which obviously leads
  to the presence of time delay between pumping steps in each curve
  caused by the same optical cycle of the incident wave. This delay
- causes a large asymmetry factor during first stage. However, as soon
- as average EHP density increases the relative contribution of this
+ causes a large asymmetry factor during the first stage. However, as soon
+ as average EHP density increases, the relative contribution of these
  pumping steps to the resulting asymmetry becomes smaller. This way
  variations of asymmetry $G_{N_e}$ synced with the period of incident
- light decreases.
+ light decrease.
 
  Higher excitation conditions are followed by larger values of
- electric field amplitude, which lead to the appearance of high EHP
+ electric field amplitude, which leads to the appearance of high EHP
  densities causing a significant change in the optical properties of
  silicon according to the equations (\ref{Index}). From the Mie
  theory, the initial (at the end of Stage~3) spatial pattern of the
@@ -666,9 +666,9 @@ GitHub~\cite{Scattnlay-web} under open source license.
  result of multipole modes superposition at the end of Stage~3 and is
  quite different as we change the size of NP. For $R=75$~nm and
  $R=100$~nm, we observe a front side asymmetry before Stage~4,
- however, the origin of it is quite different. The $R=75$~nm NP is out
+ however, its origin is quite different. The $R=75$~nm NP is out
  of resonance, moreover, Mie field pattern and the one, which comes
- from the Stage~1 are quite similar. As soon as EHP density becomes
+ from Stage~1 are quite similar. As soon as EHP density becomes
  high enough to change optical properties, the NP is still out of
  resonance, however, the presence of EHP increases absorption in
  agreement with (\ref{Index}), decreases Q-factor, and destroys
@@ -687,7 +687,7 @@ GitHub~\cite{Scattnlay-web} under open source license.
  mark.
  
  The last NP with $R=115$~nm shows the most complex behavior during
- the Stage~4. The superposition of Mie-like E-field pattern with that
+ Stage~4. The superposition of Mie-like E-field pattern with that
  from Stage~1 results in the presence of two EHP spatial maxima, back
  and front shifted. They serve as starting seeds for the EHP
  formation, and an interplay between them forms a complex behavior of
@@ -834,8 +834,8 @@ GitHub~\cite{Scattnlay-web} under open source license.
  knowledge. As a result of the presented self-consistent nonlinear
  calculations, we have obtained spatio-temporal EHP evolution inside
  the NPs and investigated the asymmetry of the EHP distributions.  We
- have revealed EHP strong asymmetric distribution during first optical
- cycles for different sizes. The highest average EHP asymmetry has
+ have revealed EHP strong asymmetric distribution during the first optical
+ cycle for different sizes. The highest average EHP asymmetry has
  been observed for NPs of smaller sizes below the first magnetic
  dipole resonance, when EHP is concentrated in the front side mostly
  during the laser pulse absorption. Essentially different EHP